A mathematician is building a new research group at Imperial College to prove, with rigorous equations, why swirling fluids like smoke plumes or ocean currents eventually settle into stable patterns or remain chaotic. This work addresses a fundamental gap in physics: the Navier-Stokes and Euler equations, which describe how fluids move, have never been fully solved for long timescales. Current theory cannot explain why some flows become turbulent while others form long-lived vortex structures, such as Jupiter’s Great Red Spot. The researcher will develop new mathematical tools from partial differential equations, harmonic analysis, and dynamical systems to tackle two core problems: how coherent structures like vortices remain stable despite fluid mixing, and how randomness influences turbulence’s statistical laws. This is fundamental science with no immediate practical application. However, rigorous proofs of fluid stability and turbulence scaling laws could eventually improve climate models that predict atmospheric jet streams, refine weather forecasting, or help design more efficient pipelines and aircraft wings. Past work in this vein has underpinned everything from satellite navigation to wind farm layout.
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The proposal focuses on the theoretical understanding of long-time dynamics questions in fluid mechanics, from the point of hydrodynamic stability and turbulence theory. By putting on rigorous mathematical grounds several classical questions, such as the meta-stability of coherent structures and the ergodic/statistical properties of fluid flows, I aim to deepen the understanding of the long- time behavior of solutions to the Navier-Stokes and the Euler equations. The main objective is to establish a research group at Imperial College to develop novel mathematical techniques in the theory of PDEs, harmonic/stochastic analysis, and dynamical systems, that would allow to move beyond the current limits of the field. These techniques will be devised and mature in the context of two fundamental problems. - Coherent structures and stability of fluids: due to its stabilizing and infinite-dimensional nature, fluid mixing causes a transfer of enstrophy to small scales, in a manner which is reversible and conservative for finite times, but results in an irreversible loss of information at infinite times. With no viscosity, it generates inviscid damping, while its interaction with diffusion creates dissipation time-scales responsible for meta-stable behavior. Rigorous results on these stability problems constitute a milestone towards the resolution of classical long-standing questions related to the transition from laminar to turbulent states and the formation of coherent vortex-like structures in large-scale atmospheric dynamics. - Cascades in stochastic fluid mechanics: the rigorous derivation of scaling laws stemming from the phenomenological theory of turbulence is a central problem in fluid mechanics. I aim to understand how randomness affects foundational aspects of the theory, such as ergodicity and anomalous dissipation, in a highly innovative context that relies on the quantification with respect to relevant parameters of classical objects such as Lyapunov exponents.
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